How to inset near a mesh boundary without creating new border faces?
You can do this using the 'Inset' tool.

Select the faces you want to inset:



Press I to trigger the 'Inset' tool and move the mouse to increase or decrease the amount. Press B during the inset to turn on the 'Boundary' option (which will stop creating faces along the boundary of the mesh): 



Press Enter when you are happy with the inset and then delete the inset faces with X> Faces